On the bidomain problem with FitzHugh–Nagumo transport
The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.
KeywordsBidomain operator Maximal \(L_p\)-regularity FitzHugh–Nagumo transport Critical spaces Global existence Stability
Mathematics Subject Classification35K50 92C35
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- 4.P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, In: Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2000.Google Scholar
- 7.M. Hieber and J. Prüss, \(L_q\)-Theory for the bidomain operator, submitted, 2018.Google Scholar
- 9.J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 1998.Google Scholar
- 10.K. Kunisch and M. Wagner, Optimal control of the bidomain system (IV): corrected proofs of the stability and regularity theorems, arXiv:1409.6904v2.
- 13.J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser, 2016.Google Scholar